Most, if not all, N contests are organised at the Metanet Forums. The two best-known of these are Nedcon and the NTTS.
Nedcon
Nedcon is a mapmaking contest, where competitors make maps based on a chosen theme. All the maps are then rated by a panel of judges, and the winning Neditor recieves a prize.
It is now Nedcon season 7, and judging is currently underway.
NTTS
The NUMA Time Trial Society, or NTTS, is a contest for serious N players. A series of maps are picked, and competitors submit demos of their two best demos on each map. These demos are then rated in two categories: Time and Best Demo. There are three places in the Time category, worth 5, 3, and 1 points. The Best Demo category has five places, worth 10, 6, 8, 4, and 2 points.
The third season of the NTTS has recently ended.
Find an article with more details of NTTS here.
Other contests
Whilst Nedcon and NTTS certainly provide plenty of contestly goodness, there are many other competitions. Some are one-offs, like the "Design N A Playpark" contest, whilst others are not, like the "Challenge of the Week" contests. Unfortunately a large percentage of these contests fail to properly get off the ground, but the "Design N A Playpark" contest is a good example of how a good idea can still result in a thoroughly successful contest. Another popular contest is The One-Life Challenge, where players compete get as far through a column as possible without dying.